Wave-mechanical momentum matrix

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

R. Benesch and V. H. Smith, Jr., Density matrix methods in X-ray scattering and momentum space calculations, in Wave Mechanics—the First Fifty Years, W. C. Price, S. S. Chissick, and T. Ravensdale, eds. (Butterworths, London, 1973), pp. 357-377. 

Bom coined the term "Quantum mechanics and in 1925 devised a system called matrix mechanics, which accounted mathematically for the posidon and momentum of the electron in the atom. He devised a technique called the Born approximation in scattering theory for computing the behavior of subatomic particles which is used in high-energy physics. Also, interpretation of the wave function for Schrodinger s wave mechanics was solved by Born who suggested that the square of the wave function could be understood as the probability of finding a particle at some point in space, For this work in quantum mechanics. Max Bom received the Nobel Prize in Physics in 1954,... 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

Generalized momentum operators as defined by Eq. (2.77) can be used in wave mechanical as well as in matrix mechanical formulations. It ensures that the operators are Hermitian, and that momenta, 7r, conjugated to generalized coordinates, qh fulfil commutation relations similar to the canonical relations of Cartesian coordinates and momenta,... 

There are situations in which a definite wave function cannot be ascribed to a photon and hence cannot quantum-mechanically be described completely. One example is a photon that has previously been scattered by an electron. A wave function exists only for the combined electron-photon system whose expansion in terms of the free photon wave functions contains the electron wave functions. The simplest case is where the photon has a definite momentum, i.e. there exists a wave function, but the polarization state cannot be specified definitely, since the coefficients depend on parameters characterizing the other system. Such a photon state is referred to as a state of partial polarization. It can be described in terms of a density matrix... 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

The branch of quantum mechanics to which we have devoted our attention in the preceding chapters, based on the Schrodinger wave equation, can be applied in the discussion of most questions which arise in physics and chemistry. It is sometimes convenient, however, to use somewhat different mathematical methods and, moreover, it has been found that a thoroughly satisfactory general theory of quantum mechanics and its physical interpretation require that a considerable extension of the simple theory be made. In the following sections we shall give a brief discussion of matrix mechanics (Sec. 51), the properties of angular momentum (Sec. 52), the uncertainty principle (Sec. 53), and transformation theory (Sec. 54). 

Thus, the adjoint relationship, expressed by the matrix G, is particularly simple. In quantum mechanics the coefficients ak have an important interpretation since they represent the amplitude of the wave function in momentum space. Equations (23) and (24) are direct analogues to the continuous Fourier transformation, which changes a coordinate... 

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